and Seo is at Department of Managerial Economics and Decision .... Though seemingly natural, the selection
SUBJECTIVE STATES: A MORE ROBUST MODEL Larry G. Epstein
Kyoungwon Seo
May 2008
Abstract We study the demand for ‡exibility and what it reveals about subjective uncertainty. As in Kreps [12], Nehring [16, 17] and Dekel, Lipman and Rustichini [5], the latter is represented by a subjective state space consisting of possible future preferences over actions to be chosen ex post. One contribution is to provide axiomatic foundations for a range of alternative hypotheses about the nature of these ex post preferences. Secondly, we establish a sense in which the subjective state space is uniquely pinned down by the ex ante ranking of (random) menus. Finally, we demonstrate the tractability of our representation by showing that it can model the two comparative notions “2 desires more ‡exibility than 1” and “2 is more averse to ‡exibility-risk than is 1.” JEL Classi…cation: D80, D81
Epstein is at Department of Economics, Boston University, Boston, MA 02215,
[email protected] and Seo is at Department of Managerial Economics and Decision Sciences, Northwestern University,
[email protected]. Epstein gratefully acknowledges the …nancial support of the NSF (award SES-0611456). We are grateful to Bart Lipman, Fabio Maccheroni and Massimo Marinacci for useful discussions and comments.
1. INTRODUCTION Following Kreps [12], Nehring [16, 17] and Dekel, Lipman and Rustichini [5] (DLR), we study the demand for ‡exibility and what it reveals about subjective uncertainty. These papers model choice under uncertainty without positing a Savage-style primitive state space. We interpret them as addressing the question: when can we view an agent choosing an action under uncertainty as though she foresees a set of possible contingencies or a subjective state space? Our principal goal is to identify those properties of (suitably de…ned) preference that justify the noted “as if” view of the agent, without otherwise restricting preference unduly, and while also permitting identi…cation of a unique subjective state space. We elaborate now on our contribution and on the value-added relative to the cited literature. Kreps studies an agent who ranks menus of actions - elements of an abstract set B - one of which is to be chosen ex post from the menu selected ex ante. When B is …nite, he shows that a simple set of axioms characterizes a representation of preference over menus (all subsets of B) that can be interpreted as re‡ecting uncertainty about future preferences over B. The representation for preference over menus that he derives has the form: Z W (x) = max v ( ) d (v) , (1.1) 2x
where x is a menu (subset of B), and is a probability measure over functions v : B ! R, each of which is a utility function representing an ex post ordering of actions. The support of can be thought of as a subjective state space underlying the ex ante ranking of menus. We think of subjective states as describing the agent’s conceptualization of the future and thus as being foreseen by her.1 (Kreps, and also Dekel, Lipman and Rustichini, consider also representations that are not additive over the possible ex post utility functions. However, in this paper we consider only additive models and when referring to the cited papers we have in mind primarily their additive models.) Two major extensions of Kreps’analysis have been pursued. One, by Dekel, Lipman and Rustichini (henceforth DLR), is motivated by the desire to derive a unique subjective state space for each agent. A …nite B does not a¤ord the richness needed to pin down the support of ; and, more generally, Kreps (Theorem 2) is 1
An alternative interpretation, developed by Kreps [14], is that these contingencies are unforeseen.
2
able only to provide a hard-to-interpret set of transformations of the subjective state space that preserve the representation of ex ante preference. DLR obtain uniqueness results by assuming that the agent (i) ex ante ranks menus of lotteries, that is, B = (B) for some …nite set of alternatives B, and (ii) satis…es alternative independence-style axioms that permit a subjective state space consisting only of vNM ex post utilities. In particular, DLR show that the subjective state space is “essentially unique”given the vNM restriction on ex post utilities. As the foundation for the existence of a subjective state space, we …nd the DLR model less than completely satisfactory. While they establish that their independence-style axioms are su¢ cient for the existence of a unique subjective state space, it seems intuitive that the perception of future contingencies does not necessarily require that preference satisfy such axioms. For example, Epstein, Marinacci and Seo [7] argue that DLR’s form of independence is violated if contingencies are coarse or ambiguous;2 the argument that independence precludes ambiguity is essentially due to Gilboa and Schmeidler [10]. But surely we would want the foundations for a subjective state space to permit the agent both to be aware that her conception of the future leaves out some relevant details (coarse states) and to be less than completely con…dent in their likelihoods (ambiguous states). Other reasons for violating independence may occur to the reader. The bottom line is that we are led to seek a model that is more robust in that it provides axiomatic foundations for a unique subjective state space while imposing less restrictive assumptions on preference. The greater robustness of our model of subjective states can be understood in part through considering the ex post utility functions v appearing in the Krepsstyle representation (1.1). In DLR, they are vNM utility functions, while in our (most general) model the v’s are required only to be upper semicontinuous - see Theorem 2.1. Since choice out of menus ex post is an integral part of the story and interpretation, if not part of the formal model, and since upper semicontinuity is the standard assumption used to ensure maximizing elements in any compact set, this suggests that our assumptions are at least close to being the minimal assumptions needed to deliver a subjective state space. (A quali…cation to minimality will be described shortly.) We adopt and extend the approach of Nehring [16, 17], who provides an al2
DLR demonstrate that a weaker (given other assumptions) form of independence, called Indi¤erence to Randomization, is also su¢ cient to establish the existence of a subjective state space. However, this axiom also precludes some forms of coarseness and ambiguity (see Epstein, Marinacci and Seo), and thus is also problematic as a foundation for subjective states.
3
ternative extension of Kreps’ model. Nehring permits the menu from which ex post choice is made to be random ex ante. In the published version, that randomness is subjective along the lines of Savage. However, in his working paper [16] he …rst considers a setting where the randomness of menus is objective - ex ante preference is over lotteries whose outcomes are menus, or over random menus. It is this version of his model that is most pertinent to our work, and thus when we refer to Nehring’s contribution, it is to his analysis for the domain of (objective) random menus. We borrow a great deal from it, including the domain of random menus for ex ante preference and our central axiom Dominance, which he calls Indirect Stochastic Dominance. In addition, Nehring [16, Section 5] points out the importance of ex post upper contour sets, which we also emphasize. (In fact, Kreps [12] was the …rst to draw attention to ex post (lower) contour sets.) We add to Nehring’s work in several ways. First, we drop his restriction that B is …nite; any compact metric space is permitted, including, in particular, the simplex (B) as in DLR. However, …nite B is also permitted, and this is noteworthy because we nevertheless provide a uniqueness result for the agent’s subjective uncertainty. This is not surprising (in light of Nehring’s analysis and) given that our domain is rich because of the presence of lotteries over menus. However, our analysis reveals more: by generalizing Nehring’s analysis to any compact metric B, we are able to show that richness of B is neither necessary (given the ranking of random menus) nor su¢ cient (in the absence of DLR’s axioms and given only the ranking of nonrandom menus) for uniqueness. In this sense, the domain of lotteries over menus of alternatives is more powerful than the DLR domain consisting of menus of lotteries over alternatives. Second, as noted, Nehring also shows the usefulness of upper contour sets for describing the uniqueness properties of representations with subjective states (see particularly his unpublished working paper). Besides generalizing his results in this regard - see Theorem 3.1 - we also elaborate and highlight this point which we feel has not been widely recognized and appreciated in the literature.3 In addition, we provide some results that have no counterparts in Nehring’s work. In particular, we generalize Nehring’s axiom (called Dominance in this paper) to a parametric family of axioms, each of which is shown to characterize, along with other more standard assumptions, a subjective state space where ex post preferences satisfy a speci…c property beyond upper semicontinuity - convex3
This may be due in part to the fact that those features of Nehring’s analysis that we emphasize appear primarily in his unpublished paper, and that the latter has also other foci the intrinsic preference for freedom of choice, for example.
4
ity is one example which is particularly important because of Epstein, Marinacci and Seo’s argument that randomization ex post can be valuable.4 In that case, the agent is sure that her preferences ex post will be convex but she is uncertain which convex preference will apply. Similarly, we provide foundations for a range of alternative hypotheses about the nature of ex post preferences, ranging from upper semicontinuity alone to the extreme of DLR’s assumptions. Epstein, Marinacci and Seo’s “second model” is another example of such an intermediate hypothesis. The capacity to incorporate such additional assumptions constitutes another sense in which our model is robust. Neither does Nehring have a counterpart of our analysis of comparative behavioral notions (described later in this introduction). Finally, we must acknowledge at the outset a limitation of our model. Though it is robust in the sense described above, this robustness comes at a cost: we assume certainty that a speci…ed element in B will be worst ex post. This assumption is needed only when B is in…nite, and then its role is purely technical - to show that one may extend a linear functional from a linear subspace to the universal in…nite dimensional linear space. Since it has no conceptual role, there is reason to hope that it might be dispensable in the future. Currently, we view it as a technical price for dealing with in…nite dimensionality. Upper Contour Sets and Uniqueness As noted, DLR prove that, under their axioms, there is an (essentially) unique representation for preference of the form (1.1) where has support on the set of vNM ex post preferences. However, as they are well aware, there may exist also other representations where ex post preferences are not vNM. Figure A.1 illustrates this possibility (dotted areas are regions of indi¤erence). In one representation, the vNM preference corresponding to v is expected with certainty, where v is normalized to have [0; 1] as its range. In the other, the agent assigns probabilities a 2 (0; 1) and (1 a) to the payo¤ functions v1 and v2 respectively, where v1 ( ) =
1 1 minfa; v ( )g and v2 ( ) = maxf0; v ( ) a 1 a
ag.
Both speci…cations imply, via (1.1), the same level of utility W (x) for any menu 4
See [18] for related results characterizing convexity of upper contour sets, albeit formulated in the context of a study of diversity rather than individual decision-making and ‡exibility.
5
x.5 Therefore, they imply the same ranking of random menus if, as assumed below, the utility of any lottery over menus is given by the expected value of W . Note that v1 and v2 do not conform to expected utility theory, but they do conform to the Betweenness axiom - ex post upper contour sets and lower contour sets are both convex - an axiom studied in risk preference theory [2, 4]. Nonuniqueness above does not rely on any special features of this example - it is the rule - and the underlying intuition for this is clear: when evaluating a given menu x ex ante, and anticipating one of fvi gni=1 to occur, the implied value of x given vi , max 2x vi ( ), depends on the highest upper contour set for vi that intersects x, and not on the entire function vi ( ). This suggests that it might be possible to piece together upper contour sets, or indi¤erence sets, from the various vi ’s to construct another 0 set fvi0 gni=1 that would lead to the same evaluation of any menu x. We see that DLR’s proposed remedy for nonuniqueness amounts to the selection of a canonical representation, consisting of vNM preferences ex post, amongst all possible representations, including those where ex post utilities may not conform to vNM. Though seemingly natural, the selection of any particular representation as canonical is invariably ad hoc. DLR o¤er two appealing justi…cations for their choice. One is minimality - they prove (Theorem 3B) that the vNM subjective state space is minimal among all subjective state spaces. Secondly, they show that their canonical representation permits an intuitive connection between the size of the state space and the desire for ‡exibility. Our concern with DLR’s treatment of uniqueness is that it is applicable given only assumptions on preference that are (for some purposes) too strong. Thus we do not adopt it here. Instead, working within the framework of preferences satisfying our weaker axioms, we propose a canonical representation that deviates from vNM ex post, but that is uniquely pinned down by preference over random menus, and that also admits a clear interpretation. We elaborate now on our approach. The key point is that, while there exist many di¤erent representations of the ex ante preference , they all induce the same (suitably de…ned) distribution m of upper contour sets (see Theorem 3.1 below).6 This is illustrated by Figure A.1. 5
Evidently, v = av1 + (1 a) v2 and v1 and v2 are comonotone, that is, v1 0 v1 ( ) v2 0 v2 ( ) 0 for all 0 and . Therefore, max 2x v ( ) = max 2x (av1 ( ) + (1 a) v2 ( )) = a max 2x v1 ( ) + (1 ) max 2x v2 ( ). 6 As indicated above, uniqueness depends on the assumption that preference is de…ned over random menus, and not merely menus as in the models of Kreps and DLR. We elaborate on this point below.
6
Recall that the range of v is [0; 1] and adopt the uniform (Lebesgue) measure on [0; 1]. Any upper contour set relevant in this example is indexed by a utility level s in [0; 1]. Thus we can identify the distribution over upper contour sets induced by v with the uniform distribution on the unit interval. Similarly, the distributions induced by v1 and v2 may be identi…ed with the uniform distributions on [0; a] and [a; 1] respectively. But the a : (1 a) mixture of these latter distributions equals the uniform distribution on [0; 1]. Thus the induced distributions over upper contour sets coincide. Since each upper contour set can be identi…ed with its indicator function, one obtains a representation of the form (1.1) where = m and each utility function v in its support is 0 1 valued. This is the (unique) canonical representation that we propose. The subjective state space consisting of (indicator functions of) upper contour sets is large - it is de…nitely not minimal in any sense. However, in addition to its main advantage - being well-de…ned given only the weak axioms speci…ed below the canonical representation we propose has the further advantage that it expresses at a glance the (unique) distribution over ex post upper contour sets implied by , and therefore also the nature of the agent’s relevant uncertainty about her ex post preferences. As an illustration, suppose that there exists one subjective state space in which all ex post preferences are convex (all upper contour sets are convex). Then uniqueness of the distribution over upper contour sets implies that for every subjective state space every ex post preference is convex; that is, convexity of ex post preferences is a feature of all subjective state spaces and thus is a property of the given ex ante preference . Therefore, the latter permits the unequivocal (independent of the representation) interpretation that the agent ranks random menus as if she is certain that all ex post preferences are convex.7 Similarly for other properties of ex post preference that can be expressed in the form “every upper contour set satis…es a suitable condition”.8 Finally, just as DLR show that the measure in their canonical representation can be used to characterize behavioral hypotheses concerning the demand for ‡exibility, Section 4 demonstrates the tractability of our representation, and 7
In contrast, if preference satis…es the DLR axioms and thus admits a representation with supported by vNM ex post utility functions, the interpretation whereby the agent is certain that she will have vNM preferences ex post is supported by one representation but not by all this is illustrated by the example in Figure A.1. 8 The collection of upper contour sets satisfying this condition must be suitably closed. Another example of such a property is Betweenness, where both upper contour sets and their complements are convex.
7
the intuitive connection that it a¤ords between subjective uncertainty and the demand for ‡exibility. We go beyond examining just the familiar notion “2 desires ‡exibility more than 1” and introduce also the new comparative behavioral notion “2 is more averse to ‡exibility-risk than is 1.”9 We show that in the DLR framework, but not in ours, 2 desires more ‡exibility than 1 if and only if 2 is more averse to ‡exibility-risk. Since these two notions seem conceptually distinct, this demonstrates another sense in which our model is more robust. The paper proceeds as follows. The next section presents our general model, that is, the model designed to capture a subjective state space and little else. Uniqueness is discussed in Section 3. Comparative behavioral notions are introduced and characterized in Section 4. Section 5 demonstrates how the general model can be specialized so as to build in a range of alternative assumptions about the expectation of ex post preferences.
2. THE GENERAL MODEL 2.1. Preliminaries Let B be a compact metric space of actions. A menu is a (nonempty) closed subset of B; K (B) denotes the set of all menus.10 A random menu is a lottery over K (B), that is, an element of (K (B)). An ex ante preference is de…ned on (K (B)). The agent ranks random menus ex ante as if expecting the following time line: a menu x is realized, then some subjective uncertainty is resolved, and …nally, at a later ex post stage she chooses an action from x. Though choice at the ex post stage is not explicitly modeled, it underlies intuition for the axioms and for the representation of . In particular, the demand for ‡exibility (the preference for large menus) is understood as arising from uncertainty about ex post preferences. A central special case is where B is a set of lotteries, B = (B) for some compact metric set B. This is the case considered by DLR, though they restrict B to be …nite, and consider preference only over (nonrandom) menus. In light of 9 Nehring [16, Section 4] de…nes a related notion of absolute risk aversion, but does not discuss or characterize comparative notions. 10 Every metric space X is endowed with the Borel -algebra, (X) denotes the set of all (Borel) probability measures on X endowed with the weak convergence topology, and K (X) is the set of all nonempty closed subsets of X endowed with the Hausdor¤ metric topology. Then (X) and K (X) are compact metric if X is compact metric. Finally, we make use of the fact that any compact metric space is also separable [1, Lemma 3.19 and Theorem 3.20].
8
the importance of this special case in the literature, and because it permits more concrete and familiar interpretations, even in the general case we sometimes refer to elements of B as lotteries. Generic elements of (K (B)) are P; P 0 ; Q; :::, generic menus are denoted x; x0 , y ..., and generic lotteries are denoted ; 0 ; , ... We make use of the fact that, by [1, Theorem 3.63], fx 2 K (B) : x open in K (B) for every open subset z B. Therefore, for any menu y, fx 2 K (B) : x \ y 6= ?g = K (B) nfx 2 K (B) : x
zg is
Bnyg
is closed, hence Borel measurable. 2.2. Axioms We adopt the following axioms for the binary relation
on
(K (B)).
Axiom 1 (Ex Ante vNM). There exists W : K (B) ! R bounded and measurable such that is represented by the expected utility function Z W (x) dP (x) . W (P ) = K(B)
The foundations for such a representation are well-known (see Fishburn [9, Theorem 10.3]). The underlying properties of preference are: completeness, transitivity, mixture continuity, independence, and an axiom, denoted A4b by Fishburn, that is similar in spirit to Savage’s P 7. The …rst four are the axioms used in the Mixture Space Theorem, and the last is needed to ensure the expected utility form. Continuity of preference is not necessary for Ex Ante vNM, though, as shown by Grandmont [11], it is su¢ cient when combined with completeness, transitivity and independence. (See Kreps [13, pp. 59-67] for a textbook discussion.) Because we criticized DLR’s adoption of independence in the introduction, it is important to distinguish DLR’s version of independence from that implied by Ex Ante vNM. The latter version has the following form: For all random menus P; P 0 and Q and for all 0 < < 1, P0
P () P 0 + (1
)Q
9
P + (1
) Q:
To interpret this condition, note that, since a mixture such as P + (1 ) Q is a random menu, it follows from the time line described above that a speci…c menu is realized before the agent sees a subjective state and chooses from the menu. In particular, therefore, all randomization in both component measures P and Q, as well as in the mixing is completed before then. It is this immediacy of the randomization that renders this version of independence intuitive and that distinguishes it from DLR’s version, where the coin toss corresponding to the mixing is completed after choice from the menu. To see why the timing of randomization can matter, think of an ambiguity averse agent as imagining herself playing a game against a malevolent nature. She suspects that, after she has chosen an action ex post out of the available menu, nature will choose a probability law over the remaining uncertainty in a way that is unfavorable for her. Then, randomization that is completed immediately, before nature acts, does nothing to impede persecution by nature. In contrast, randomization that is conducted after nature moves, as in DLR’s form of independence, can be bene…cial because it places nature at a disadvantage.11 Though we do not assume that preference is continuous, we do assume that it satis…es the following weaker requirement.12 Axiom 2 (Right-Continuity). If xn & x and xn
y for all n, then x
By xn & x, we mean the set-theoretic conditions xn+1 Note, however, that for a declining sequence,
y.
xn and \1 1 xn = x.
\1 1 xn = x if and only if lim xn = x, where the latter indicates convergence in the Hausdor¤ metric.13 Consequently, the axiom is weaker than upper semicontinuity on K (B), that is, all sets of the form fx 2 K (B) : x yg being closed. The next axiom excludes total indi¤erence. Axiom 3 (Nondegeneracy). There exist random menus such that P 11
P 0.
See [7] for elaboration. We adopt the obvious notation, whereby x is identi…ed with x and so on. 13 Apply the characterization of Hausdor¤ convergence [1, Theorem 3.65]: let \1 1 xn = x. c c Then x xn =) x Lixn . Also, 62 x =) 62 xN =) 2 G (xN ) ([1 N xn ) for some N and open set G =) 62 Lsxn . Conclude that x Lixn Lsxn x, which implies Lixn = Lsxn = x, and hence xn ! x. The converse is also straightforward. 12
10
Our key axiom is the translation into our setting of Nehring’s axiom (Indirect Stochastic Dominance). For any two random menus P 0 and P , say that P 0 dominates P , if, for all “relevant”menus y, P 0 (fx 2 K (B) : x \ y 6= ?g)
P (fx 2 K (B) : x \ y 6= ?g) .
(2.1)
To interpret (2.1), think of y as an upper contour set for some conceivable ex post preference over actions. Thus actions in y are “desirable” according to that ex post preference and x \ y 6= ; indicates that x contains at least one desirable action, in which case we might refer to x as being desirable. Accordingly, P 0 dominates P if the probability of the realization of a desirable menu is larger under P 0 , and if this is true for every set y and hence for every conceivable de…nition of “desirable.” It remains to specify what is a “relevant” menu. We take it to mean that y 2 K , where K = fy 2 K (B) : 62 yg , for some …xed in B. That is, P 0 dominates P if (2.1) is valid for all y in K . Think of as an action for which there is ex ante certainty that it will worst ex post. Then upper contour sets that include must include all actions and thus may be safely ignored - intuitively, in order to de…ne a meaningful notion of ‘desirable’, an upper contour set should exclude something. The assumption of ex ante certainty that a speci…ed action will be worst ex post was mentioned in the introduction, where it was pointed out that it is needed only in the case where B is in…nite. Formally, when B is …nite, all results below remain valid if dominance is de…ned instead by requiring (2.1) to be satis…ed for all y in K (B).14 Axiom 4 (Dominance). If P 0 dominates P , then P 0
P.
There are two notable implications of Dominance. First, when P 0 = x0 and P = x are degenerate, then x0 dominates x if x0 x. Therefore, Dominance implies Monotonicity: x0 x =) x0 x. 14 The existence of the worst lottery plays a role only in the proof of Theorem 5.1, which generalizes Theorem 2.1. The remark following Lemma A.5 describes how it can be dispensed with when B is …nite.
11
It also implies (given Ex Ante vNM) Kreps’second key axiom [12, condition (1.5)]: given any menus x; x1 and x2 , let P0 =
1 2 x[x1
+
1 2 x[x2
and P =
1 2 x
+
1 . 2 x[x1 [x2
(2.2)
Then P 0 dominates P , and thus Dominance implies that 1 2 x[x1
+
1 2 x[x2
1 2 x
+
1 . 2 x[x1 [x2
Deduce (from independence) that x
x[x1
=)
x[x1 [x2 .
x[x2
Since Monotonicity is also implied, we have …nally that (in friendlier notation) x
x [ x1 =) x [ x2
x [ x2 [ x1 ,
which is Kreps’axiom. 2.3. Utility Think of minimal assumptions on the nature of ex post preferences over B. Completeness and transitivity are relatively innocuous. In order to ensure the existence of optimal elements in every menu ex post (though ex post choice exists only in the mind of the agent), suppose that ex post preferences are upper semicontinuous. Since B is compact metric, every such ex post preference can be represented by an upper semicontinuous payo¤ function (in fact, this is true much more generally - see Rader [20], for example). It follows that any upper semicontinuous ex post preference that is not total indi¤erence and that ranks the speci…ed lottery as worst has a utility representation by some (nonunique) v : B ! R lying in V - the set of all upper semicontinuous (ex post) payo¤ functions satisfying 0 = v(
)
v( )
max v ( ) = 1: 2B
(2.3)
We need a topology (and corresponding Borel -algebra) for V that we now describe. Denote by U SC (B) the set of upper semicontinuous (usc) functions from B into [0; 1]. Adopt the topology generated by the subbasis: fv : sup v ( ) > g and fv : sup v ( ) < g 2z
2x
12
where z and x vary over open and compact (or equivalently, closed) sets respectively. This is the weakest topology such that the mapping v 7 ! sup 2y v ( ) is lower semicontinuous (lsc) for each open y and usc for each closed y. Then renders U SC (B) compact metric and V is closed in U SC (B). See [19] for details about supporting assertions made here; for an application of this topology in economics, and for other properties, see Epstein and Peters [8].15 A critical property of is that it is consistent with the Hausdor¤ metric topology on K (B). Each closed subset y can be identi…ed with the usc function 1y ( ). Under this identi…cation, K (B) U SC (B) and the restriction of coincides with the Hausdor¤ metric topology. Any Borel probability measure 2 (V ) generates a utility function W on (K (B)) of the form:16 Z Z W (P ) = max v ( ) d (v) dP (x) . (2.4) 2x
Refer to as a representation (of the preference corresponding to W). The next theorem is our …rst main result. It is implied by Theorem 5.1, whose proof is provided in the appendix. Theorem 2.1. satis…es Ex Ante vNM, Right-Continuity, Nondegeneracy and Dominance if and only if it admits a representation. The implied utility for (nonrandom) menus is W : K(B) ! R, where W has the form described in the introduction: Z W (x) = max v ( ) d (v) . V
2x
As described earlier, Kreps [12] derives such a representation when B is …nite, and DLR characterize the special case where B is the simplex and has support on the set of vNM utility functions. Nehring [16, Theorem 1] proves a counterpart of Theorem 2.1 without requiring a worst action, but under the assumption that B 15
One such property, used below, is that the mapping (v; ) 7 ! v ( ) is usc on U SC (B) B. An implication is that (x; v) 7 ! max 2x v ( ) is usc; this follows from a form of the Maximum Theorem [1, Lemma 14.29]. 16 The integral is well-de…ned by the Fubini Theorem because (x; v) 7 ! max 2x v ( ) is usc, hence product measurable.
13
is …nite.17 In our more general setting, upper semicontinuity of ex post preference (or utility) is of interest, and it is characterized in our theorem. Observe that the utility function in (2.4) is upper semicontinuous on (K (B)).18 Two implications follow. First, not only is (hypothetical or notional) ex post choice out of menus well-de…ned, but so also is the (‘real’ or part of the formal model) ex ante choice out of any compact feasible set of random menus. Secondly, since our axioms characterize the functional form, they necessarily imply upper semicontinuity of preference.
3. WHAT IS REVEALED BY THE RANKING OF RANDOM MENUS? We have seen that the representing measure provided by Theorem 2.1, and more to the point even its support, are not unique. Before describing what is uniquely determined by preference, it is useful to consider …rst the reasons for nonuniqueness. One reason that comes to mind is the state-dependence problem one can always rescale each v ( ) by a positive multiplicative constant av and then use the modi…ed measure d 0 = d =av . Such rescaling does not a¤ect the support and besides it is ruled out by the normalizations of payo¤ functions in (2.3). Therefore, nonuniqueness arises here for another reason. We are conditioned to feel that preference ‘should’reveal beliefs byRSavage’s celebrated theorem. However, think of the functional form W (x) = V max 2x v ( ) d (v) as the subjective expected utility of the (real-valued) act fx , fx (v) = max 2x v ( ), where the state space is V . Savage is able to determine a unique probability measure only by assuming that the agent ranks all acts over the state space. But here, the relevant set of acts ffx : x 2 K (B)g is only a ‘small’proper subset (in fact, every fx is usc and increasing in the pointwise ordering on V ). Thus one should not expect observable choice to determine a unique probability measure over states, and the focus should be rather to identify what is in fact pinned down by observable choice. As illustrated by Figure A.1 and the surrounding discussion, intuition suggests that only the upper contour sets associated with ex post preferences, and not 17 Nehring [17, p. 108] asserts that his representation theorem generalizes to the in…nite case, and gives a very brief and incomplete sketch of how to achieve it. Our approach is di¤erent. 18 By [15, Proposition D.7], it is monotone and right-continuous. Secondly, R W ( ) is usc because R if Pn ! P , then lim sup W (x) dPn WR(x) dP by the nature of the weak convergence topology [1, Theorem 12.4]. Therefore, P 7 ! W (x) dP is usc.
14
the latter per se, matter for ex ante choice. Accordingly, we will show that the ranking of random menus pins down, and is in turn completely determined by, the (suitably de…ned) distribution over ex post upper contour sets that is implied by any representing . This will be shown to provide another perspective on and rationale for our choice of a canonical representation (see the discussion in the introduction). For each 2 (V ), de…ne a Borel probability measure m 2 (K (B)), viewed as a measure over upper contour sets. Let Ts : V ! K (B) be given by Ts (v) = f : v ( )
sg , for each s 2 [0; 1],
(3.1)
and de…ne m ( ) , on the Borel -algebra, by m ()=
Z
1
Ts 1 ( ) dL (s) ,
(3.2)
0
where dL denotes the Lebesgue measure on [0; 1]. It is straightforward to see that m is a well-de…ned measure: de…ne to be the collection of all Borel sets A such that s 7! Ts 1 (A) is Lebesgue integrable. Because is countably additive, is a -algebra. Moreover, contains all sets of the form fy 2 K (B) : x \ y 6= ?g for x 2 K (B) : (3.3) This is because Ts 1 (fy : x \ y 6= ?g) = fv 2 V : x \ f : v ( ) =
v 2 V : max v ( ) 2x
sg = 6 ?g
s ,
which is Borel measurable because it is -closed, and because Ts 1 (fy : z \ y 6= ?g) is nonincreasing on [0; 1], and thus (Riemann) integrable. But the Borel -algebra is the smallest -algebra containing all sets of the form (3.3) - see [1, Theorem 14.69] - and thus contains all Borel sets. One can interpret m as summarizing the probability distribution of upper contour sets generated by a 2–stage process. First, a utility level s is drawn from a uniform distribution over [0; 1], and then an ex post utility function v, (and thus also the upper contour set Ts (v)), is drawn according to . In short, m is the “expected distribution”over upper contour sets induced by . 15
A simple example may be useful. Let y1 v ( ) = s1 1y1 ( ) + (1
s1 )1y2 ( ) =
y2 8 < :
B, where if
1 1
62 y2 , and de…ne
s1 0
2 y1 2 y2 ny1 otherwise.
Then v lies in V and has upper contour sets y1 , y2 and B. Setting = representation for the preference with utility function Z W (P ) = max (s1 1y1 ( ) + (1 s1 )1y2 ( )) dP (x) . K(B)
v
gives a
2x
The ‘natural’interpretation is that of certainty that the ex post payo¤ function will be v. Compute that m has two points of support: m (fy1 g) = s1 , and m (fy2 g) = (1
s1 ) .
(3.4)
The third upper contour set B receives no weight, re‡ecting the fact that it is common to all payo¤ functions in V and thus is not relevant to distinguishing our particular v. But there is another representation 0 for the same preference: let 0 assign probability s1 to v1 and (1 s1 ) to v2 , where vi ( ) = 1yi ( ), i = 1; 2. Then 0 de…nes, via (2.4), the same utility function W given above, because max (s1 1y1 ( ) + (1 2x
s1 )1y2 ( )) = s1 max 1y1 ( ) + (1 2x
s1 ) max 1y2 ( ) , 2x
though it suggests a di¤erent interpretation - uncertainty about whether payo¤s will be given by v1 or by v2 . Note, however, that 0 and have in common the induced distribution over upper contour sets, that is, m 0 = m . The uniqueness of this induced distribution across all representations is established more generally in the next theorem. A possibly puzzling feature of the example is that the measure m de…ned in (3.4) involves the utility levels s1 and s2 . It is important to keep in mind, however, that these are not ordinal values, but rather have unambiguous meaning in terms of the given preference over random menus. For example, for the preference order in the example, s1 is the unique probability p such that the random menu (B; (1 p); f g; p) is indi¤erent to receiving the menu f g with certainty, where is any lottery in y2 ny1 . This illustrates that the adoption of a domain of random menus is crucial for our analysis (another illustration is given below). The main result of this section is that any two representations for our axioms) generate the identical measure over upper contour sets. 16
(satisfying
Theorem 3.1. Suppose that Then: (a) For all x 2 K (B),
satis…es our axioms and that
m (fy 2 K (B) : x \ y 6= ?g) = (b) Let
0
be any other representation for
Z
is a representation.
max v ( ) d (v) .
V
2x
(3.5)
. Then m 0 = m .
Note that the ranking of (nonrandom) menus alone is not su¢ cient to pin down a unique measure on upper contour sets. For example, let y; y 0 be closed subsets, and let 1 1 (3.6) y + y 0 and 2 2 1 1 1 m1 = y + y0 + y[y0 : 3 3 3 Then m1 and m2 represent the same preference on K (B), via (3.5), but they imply di¤erent rankings on (K (B)). This is easy to see: for i = 1; 2, let Wi (x) = mi (fy 2 K (B) : x \ y 6= ?g). On the domain of nonrandom menus, W2 assumes the values 0; 12 or 1 depending on whether the menu in question intersects none, one or both of y and y 0 , while W1 assumes the values 0; 32 or 1 in the same circumstances. Therefore, they are ordinally, but not cardinally, equivalent on K (B). m2 =
Proof: (a) Compute that m (fy : x \ y 6= ?g) = =
Z
1
Ts 1 (fy : x \ y 6= ?g) ds
0
Z
1
(fv : x \ f : v ( )
0
=
Z
0
sg = 6 ?g) ds
1
v : max v ( ) Z
2x
1
s
ds
= 1 v : max v ( ) < s ds 2x 0 Z 1 = sdF (s) , (F (s) = v : max v ( ) < s 2x 0 Z = max v ( ) d (v) , V
2x
17
),
where the next to last equality follows from integration by parts, and the last by a change of variables. (b) If and 0 are any two representations, then can R be represented as an expected utility function with vNM Rindex W , W (x) = max 2x v ( ) d (v), and also with vNM index W 0 , W 0 (x) = max 2x v ( ) d 0 (v). By the uniqueness properties of vNM utility, it follows that, for some a > 0 and b 2 R, and for all x 2 K (B), Z Z max v ( ) d 0 (v) = a
max v ( ) d (v) + b.
2x
Letting x = f Z
2x
g and x = B yields 0 = b and 1 = a + b, which implies Z 0 max v ( ) d (v) = max v ( ) d (v) for all x 2 K (B) : 2x
2x
Hence, by (a), m 0 (fy 2 K (B) : x \ y 6= ?g) = m (fy 2 K (B) : x \ y 6= ?g) for all x 2 K (B) : But these sets generate the Borel -algebra [1, Theorem 14.69] - hence m 0 = m . The theorem proves not only that the implied distribution over ex post upper contour sets is unique (part (b)), but also that it contains all relevant information about preference - indeed, by (a), we can rewrite the utility function W from (2.4) in the form Z W (P ) = m (fy : x \ y 6= ?g) dP (x) . (3.7) Evidently,
m (fy : x \ y 6= ?g) =
Z
max 1y ( ) dm (y) : 2x
(3.8)
Since each indicator function 1y ( ) is usc, indeed an element of V , and since K (B) is homeomorphic to a subspace of U SC (B), m can be viewed as a representation. This perspective on the theorem relates more explicitly to the discussion in the opening paragraph of this section concerning a unique canonical representation. The expression (3.8) suggests that uncertainty about ex post preferences is con…ned to binary utility functions. Though binary payo¤ functions are clearly very special, we feel that nevertheless the representation (3.8) is useful as a reduced form: the agent may view more general (nonbinary) payo¤ functions as possible, 18
but, as we have seen, it is only the implied uncertainty about upper contour sets that matter for the ranking of random menus. Thus ultimately all that matters for observable behavior are these expectations regarding upper contour sets, or equivalently their indicator functions, which are captured by the representation (3.8). Another implication of the theorem is worth emphasizing: equation (3.2) describes all the representations corresponding to a …xed preference. Let satisfy our axioms. Then there exists a unique canonical representation m as in (3.7). Now let be any measure satisfying, on the Borel -algebra, Z 1 m( ) = Ts 1 ( ) dL (s) . 0
Then represents, via (2.4), some preference 0 over random menus. But m de…ned by (3.2) also represents 0 . However, m = m and thus 0 = . In other words, represents if and only if it satis…es (3.2). Finally, there is a sense in which the uniqueness proven in the theorem is not completely satisfactory. The theorem shows that every representation generates the same measure m over upper contour sets, but the de…nition of “representation” imposes a priori the restriction (2.3), whereby is the worst alternative for all ex post payo¤ functions and the latter are normalized or scaled so that v ( ) = 0 and max 2B v ( ) = 1. We conclude this section by describing how the uniqueness property of m is modi…ed when the de…nition of “representation”is broadened to include any probability measure 0 on U SC (B) such that (2.4) is a utility function for . De…ne m 0 by the counterpart of (3.2), Z 1 0 m 0 (A) = (fv 2 U SC (B) : f : v ( ) sg 2 Ag) dL (s) . 0
Then m 0 is a positive Borel measure satisfying m 0 (K (B)) 1; equality with 1 does not obtain in general because f : v ( ) sg can be empty with positive probability according to the product measure 0 L on U SC (B) [0; 1].19 Let be a representation as provided by Theorem 2.1, and let m be the probability measure over upper contour sets that it induces. To relate m 0 and m , argue as in the proof of part (b) of Theorem 3.1 that m 0 (fy 2 K (B) : x \ y 6= ?g) = am (fy 2 K (B) : x \ y 6= ?g) + b; 19
m 0 (K (B)) = 1 if and only if
0
(fv 2 U SC (B) : max
19
2B
v ( ) = 1g) = 1.
for some a > 0 and b 2 R, and for all x 2 K (B). Setting x = B yields a + b Note that
1.
m 0 (fy 2 K (B) : x \ y 6= ?g) = am (fy 2 K (B) : x \ y 6= ?g) + b = (am + b B ) (fy 2 K (B) : x \ y 6= ?g) : Since these sets generate the Borel -algebra [1, Theorem 14.69], m 0 = am + b B : Thus while the two induced measures m 0 and m are in general distinct, with m 0 not even being a probability measure, they agree once conditioned on K (B) nfBg. Call an upper contour set proper if it is not equal to B. Conclude that all (broadly de…ned) representations induce the identical distributions over the set of proper upper contour sets. In particular, uniqueness of the distribution induced over proper upper contour sets is una¤ected even by the well-known state-dependence problem.
4. FLEXIBILITY AND FLEXIBILITY-RISK Here we show that our model is useful for capturing intuitive forms of behavior having to do with ‡exibility. A limitation of the DLR model in this regard is pointed out, thereby establishing another sense in which our model is more robust. Adopt the following notation: for the vNM index W provided by Ex Ante vNM, de…ne W (x [ x1 ) ; and x1 W (x) = W (x) x2
x1 W
(x) =
W (x [ x1 )
W (x)
x1 W
(x)
x1 W
(W (x [ x2 )
(x [ x2 ) =
W (x [ x1 [ x2 )) .
For later reference, de…ne also, for every n > 1, xn :::
x1 W
(x) =
xn
1
:::
x1 W
(x)
xn
1
:::
x1 W
(x [ xn ) :
(4.1)
Given the canonical representation m provided by Theorem 3.1, we have: W (x) = m (fy : y \ x 6= ?g) ; x1 W
(x) = m (fy : y \ x = ?; y \ x1 6= ?g) , and 20
x2
x1 W
(x) = m (fy : y \ x = ?; y \ x1 6= ?; y \ x2 6= ?g) .
(4.2)
For any agent satisfying our axioms, larger menus are weakly preferred, which we describe in terms of a demand for (or value of) ‡exibility. This demand may be characterized simply, since x [ x1 x () x1 W (x) > 0 and so x [ x1
x () m (fy : y \ x = ?; y \ x1 6= ?g) > 0.
(4.3)
Thus the value of ‡exibility is summarized by properties of m on 1 , the collection of all sets of the form fy : y \ x = ?; y \ x1 6= ?g as x and x1 vary over all menus. Now compare the desire for ‡exibility of two agents. Let 1 and 2 be two preferences satisfying our axioms (with representing measures m1 and m2 ). Say that 2 desires ‡exibility more than 1 if x [ x1
1
x =) x [ x1
2
x,
that is, if whenever 1 strictly values the ‡exibility a¤orded by x1 nx, then so does 2. Then it follows from (4.3) that 2 desires more ‡exibility than 1 if and only if m1 is absolutely continuous with respect to m2 on 1 (abbreviated m1 maxfr; r g. Then P = a Q and P 0 = a Q0 lie in and = a(P P 0 ) . Extend W 0 to W 1 on W1 (
by linearity:
) = rW (P ) R or equivalently, W 1 ( ) = W d .
r0 W (P 0 ) , if
Lemma A.4. W 1 is a positive linear functional on
32
= rP
.
r0 P 0 ,
Proof : To show that W 1 is linear, note that Z 1 0 1 0) W ( + = W W d ( + 0 0) + 0 0 = Z Z 0 = Wd + W d 0 = W1 ( ) +
0
W 1 ( 0) .
Now show that 0 =) W 1 ( ) 0: By Lemma A.3(iv), = a(P P 0 ) , 0 and thus 0 =) P =) W (P ) W (P ), by Lemma A.2(i) and 0 P 1 0 Y -Dominance. Thus W ( ) = a (W (P ) W (P )) 0. Lemma A.5. W (x) =
R
V
max
2x
v ( ) d (v) for some
2 ba1+ V Y .
Proof : Note that Bb V Y is a Riesz space with unit 1 and is a vector subspace containing 1. Thus the positive linear functional W 1 on admits a positive linear c to Bb V Y [1, Corollary 6.32]. By the Riesz Representation Theorem extension W [6, IV.5.1], there exists a Borel charge 2 ba V Y such that Z c( ) = W (v) d (v) . VY
c (1F ) for any measurable subset F of V Y and since W c is positive, Since (F ) = W Y we have 2 ba+ V : Consequently, Z c W (x) = W ( x ) = W = max v ( ) d (v) : x VY
2x
Recall that W (B) = 1 and max 2B v ( ) = 1 for each v 2 V Y . Thus Z Z Y V = 1d (v) = max v ( ) d (v) = W (B) = 1. VY
VY
2B
Remark 1. When B is …nite, we can dispense with . Modify the above proof R as follows. Let (y) = (max 2x 1y ( ) 1) d (x) for each y 2 Y and 2 ca (K (B)). Then, Lemmas A.3-A.5 are still true when we replace P = f g by P = B and V Y by Y . We lose the unit 1, but since B is …nite, so is Y and we can extend W 1 to Bb (Y ) in the proof of Lemma A.5. The rest of the proof is unchanged. 33
The rest of the proof consists of invoking the Choquet Theorem to get a Borel measure on K (B), which turns out to be a Y -representation. Lemma A.6. For any 2 ba1+ (V ) satisfying A.2, there exists a unique Borel probability measure m on K (B) such that, for every x, Z W (x) max v ( ) d (v) = m (fy 2 K (B) : x \ y 6= ?g) . (A.3) V
2x
Proof : W is right-continuous, that is, if xn & x, then W (xn ) & W (x) . In addition, W is completely alternating, that is, xn :::
x1 W
(x)
0,
for every n 1 and x; x1 ; :::; xn 2 K (B); recall (4.1). Here is a veri…cation: because x 7! max 2x v ( ) is completely alternating for usc v [15, p.11], we have, Z d (v) 0. xn ::: x1 W (x) = xn ::: x1 max v ( ) 2x
As noted earlier, since B is compact metric, it is also separable. Therefore, by the Choquet Theorem [15, Theorem 1.13], there exists a unique measure m satisfying (A.3), de…ned on the Borel -algebra generated by the Fell topology on K (B). Since B is compact metric, the Fell topology is equivalent to the Hausdor¤ metric topology [1, Section 3.17]. This completes the proof. Remark 2. Following common terminology in the theory of capacities, W is in…nitely alternating if ! ! n \ X [ jIj+1 W xi ( 1) W xi , i=1
fI:?6=I f1;:::;ngg
i2I
for all n 2 and x1 ; :::; xn 2 K (B). It is straightforward to show that W is completely alternating if and only if it is monotone and in…nitely alternating. R Lemma A.7. Suppose on (K (B)) is represented by W (P ) = K(B) W (x) dP (x) and (A.3). If satis…es Y -Dominance, then m (Y ) = 1.26 26
Note that Y2 is used heavily in the proof.
34
Proof : Step 1. We show that m (fy 2 K (B) : x \ y = ?; x1 \ y 6= ?; :::; xk \ y 6= ?g) = 0,
(A.4)
for all (nonempty) closed subsets x; x1 ; :::; xk of B, such that fy 2 K (B) : x \ y = ?; x1 \ y 6= ?; :::; xk \ y 6= ?g
K (B) nY .
(A.5)
Consider x; x1 ; :::; xk 2 K (B). Assume (A.5) and note that m (fy : x \ y = ?; x1 \ y 6= ?; :::; xk \ y 6= ?g) X X S S = W x [ xi W x [ xi i2I
I f1;:::;kg jIj=1;3;:::
= 2k
1
(W (P )
i2I
I f1;:::;kg jIj=0;2;:::
W (P 0 ))
where P =
X
1
I f1;:::;kg jIj=1;3;:::
2k
1
x[
S
i2I
xi
!
and P 0 =
X
1
I f1;:::;kg jIj=0;2;:::
It is clear that P; P 0 2 (K (B)). Next we prove P W (P ) = W (P 0 ) by Y -Dominance, and (A.4) follows. Take any y 2 Y and show that
Y
2k
1
x[
S
xi
i2I
P 0 and P 0
P (fx 2 K (B) : x \ y 6= ?g) = P 0 (fx 2 K (B) : x \ y 6= ?g) :
!:
Y
P . Then
(A.6)
If x \ y 6= ?, then both sides equal 1 and thus (A.6) is clear. In light of (A.5), it remains only to consider the case where x \ y = ? and xj \ y = ? for some j = 1; :::; k. But then x[
S
i2I
xi \ y 6= ? , x [
and (A.6) is implied.
S
i2Infjg
xi \ y 6= ?;
Step 2. The assertion in Step 1 extends to all x 2 K (B) and all open subsets x1 ; :::; xk of B: Since B is metrizable, there are sequences zi;n of closed sets such that zi;n % xi for each i = 1; :::; k. Apply Step 1 to fzi;n gi=1;:::;k for each n, let n go to in…nity, and apply the countable additivity of m. 35
Step 3. Complete the proof of the lemma. De…ne K = fy 2 K (B) : Then K and Y [ K are both closed. The collection of sets of the form
2 yg.
fy 2 K (B) : x \ y = ?; x1 \ y 6= ?; :::; xk \ y 6= ?g , for closed x and open x1 ; :::; xk , constitutes a base for the Hausdor¤ metric topology - see [1, Lemma 3.66]. Since K (B) is metrizable, the open set K (B) n (Y [ K ) is a countable union of basic sets. Thus there exist basic open subsets An of K (B) such that 1 S
m (K (B) n (Y [ K )) = m
n=1
An
1 X
m (An ) = 0;
n=1
equality with zero follows from Step 2 and the inclusions An K (B) nY . Finally, 1
K (B) n (Y [ K )
m (Y ) = m (Y [ K ) m (K ) = 1 m (fy 2 K (B) : f g \ y 6= ?g) = 1 W f g = 1:
Since Y is embedded in V Y by the identi…cation y 7! 1y , m in the previous Lemma can be viewed as an element of V Y and hence we have a Y representation. Proof of Theorem 5.1(b): By the argument at the very end of Section 5, 1 = R1 0 1 m 0 (Y ) = 0 Ts (Y ) dL (s). Therefore, 0 Ts 1 (Y ) = 1 for all s 2 E (0; 1], where L (E) = 1. There exists a countable subset E of E that is dense in (0; 1]. (The open intervals can be enumerated fIn g. For every open interval In , we can pick en 2 In \ E - the intersection must be nonempty. Let E = fen g.) Since 0 is countably additive, 0 \s2E Ts 1 (Y ) = 1.
But \s2(0;1] Ts 1 (Y ) = \s2E Ts 1 (Y ). (See the proof of (A.1); the latter refers to the special case where E is the set of rationals, but only the denseness of E is important.) Therefore, 0 \s2(0;1] Ts 1 (Y ) = 1. Finally, note that V Y = \s2(0;1] Ts 1 (Y ).
36
References [1] C.D. Aliprantis and K.C. Border, In…nite Dimensional Analysis, Springer, 1994. [2] S.H. Chew, A generalization of the quasilinear mean with applications to the measurement of income inequality and decision theory resolving the Allais Paradox, Econometrica 51 (1983), 1065-1092. [3] G. Choquet, Theory of capacities, Annales de l’Institut Fourier 5 (1953), 131-295. [4] E. Dekel, An axiomatic characterization of preferences under uncertainty, J. Econ. Theory 40 (1986), 304-318. [5] E. Dekel, B. Lipman and A. Rustichini, Representing preferences with a unique subjective state space, Econometrica 69 (2001), 891-934. [6] N. Dunford and J.T. Schwartz, Linear Operators, Part I: General Theory, Wiley, 1988. [7] L.G. Epstein, M. Marinacci and K. Seo, Coarse contingencies and ambiguity, Theoretical Econ. 2 (2007), 355-394. [8] L.G. Epstein and M. Peters, A revelation principle for competing mechanisms, J. Econ. Theory 88 (1999), 119-160. [9] P. Fishburn, Utility Theory for Decision Making, John Wiley, 1970. [10] I. Gilboa and D. Schmeidler, Maxmin expected utility with non-unique priors, J. Math. Econ. 18 (1989), 141-153. [11] J. M. Grandmont, Continuity properties of a von Neumann-Morgenstern utility, J. Econ. Theory 4 (1972), 45-57. [12] D. Kreps, A representation theorem for ‘preference for ‡exibility’, Econometrica 47 (1979), 565-577. [13] D. Kreps, Notes on the Theory of Choice, Westview, 1988. [14] D. Kreps, Static choice in the presence of unforeseen contingencies, in Essays in Honour of F. Hahn, P. Dasgupta et al eds., MIT Press, 1992. 37
[15] I. Molchanov, Theory of Random Sets, Springer, 2005. [16] K. Nehring, Preference for ‡exibility and freedom of choice in a Savage framework, UC Davis Working Paper, 1996. [17] K. Nehring, Preference for ‡exibility in a Savage framework, Econometrica 67 (1999), 101-119. [18] K. Nehring, Diversity and the geometry of similarity, 1999. [19] T. Norberg, Random capacities and their distributions, Probab. Th. Rel. Fields 73 (1986), 281-297. [20] T. Rader, The existence of a utility function to represent preferences, Rev. Ec. Stud. 30 (1963), 229-232.
38
{β : v(β ) = a}
v, probability 1
{β : v1 (β ) = 1}
1 min{a,v(β )}, a probability a
v1 =
{β : v2 (β ) = 0}
1 max{0, v(β )− a}, 1− a probability 1 − a
v2 =
Figure A.1: Two subjective state spaces
39
Ts (v ) = {β : v(β )≥ s}
{β : v (β )= v (β ')} *
β' β ''
{
*
x ε = β : v * (β )≤ v* (β ')− ε
γ Figure A.2: Proof of of Theorem 4.1
40
}
